open sets and Borel sets on the extended real line

Question

I know the definition of an open set and a Borel set on the real line $\mathbb R$:

(1) $A \subset \mathbb R$ is open if every $x \in A$ is an interior point, i.e. there exists some $\delta>0$ s.t. $y\in A$ whenever $|y-x|<\delta$.
(2) The collection $\mathcal B$ of Borel sets is generated by the class of open sets in $\mathbb R$.

I'm not sure how these two definitions should be modified when we work with the extended real line, i.e. $\overline {\mathbb R}=\mathbb R \cup \{-\infty, \infty\}$?

In particular, consider, for instance, the openness of $A\triangleq(1, \infty]$. Obviously every $x\in (1, \infty)$ is an interior point of $A$, according to the definition above. But what about $x=\infty$? Clearly no real number $y$ would be within any finite distance from $\infty$, and $\infty-\infty$ is undefined. So it's not clear to me if and how $\infty$ should be considered an interior point of $A$?

Is the collection $\overline {\mathcal B}$ of Borel sets in $\overline{\mathbb R}$ still defined as the $\sigma$-field generated by open sets in $\overline{\mathbb R}$?

I know only the basics of analysis and measure theory. I'd appreciate it if someone can help me out here. Thanks a lot!

Answer 1

The extended real line is homeomorphic to the interval $C=[0,1]$ with the usual topology inherited from the real line.

So, you know all the open sets in the extended real line by knowing the open sets on $C$.

As to the neighborhoods of $\infty$, this quote from the Wiki page explains (note the last line).

In this topology, a set U is a neighborhood of $\infty$ if and only if it contains a set $\{x : x > a\}$ for some real number a, and analogously for the neighborhoods of $-\infty$. The extended real line is a compact Hausdorff space homeomorphic to the unit interval [0,1]. Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval. **There is no metric that is an extension of the ordinary metric on $\mathbb{R}**$.

And yes, the Borel sets are still the $\sigma$-field generated by these open sets.

Edit: I would like to add from DanielWainFleet’s excellent characterization below, pointing out that $B$ is a Borel set in the extended reals if $B\setminus\{\infty, -\infty\}$ is a Borel set in the reals.